Semiclassical Szego limit of resonance clusters for the hydrogen atom Stark Hamiltonian

TL;DR

This paper extends Szego-type theorems to resonance clusters of the hydrogen atom under a Stark field, revealing that resonance shifts concentrate on classical energy surfaces in the weak field limit.

Contribution

It introduces a semiclassical Szego-type theorem for resonance clusters involving non self-adjoint operators and unbounded Stark perturbations.

Findings

Resonance shifts concentrate on classical energy surfaces.

Extension of Szego-type results to resonance clusters.

Use of localization properties of coherent states for control.

Abstract

We study the weighted averages of resonance clusters for the hydrogen atom with a Stark electric field in the weak field limit. We prove a semiclassical Szego-type theorem for resonance clusters showing that the limiting distribution of the resonance shifts concentrates on the classical energy surface corresponding to a rescaled eigenvalue of the hydrogen atom Hamiltonian. This result extends Szego-type results on eigenvalue clusters to resonance clusters. There are two new features in this work: first, the study of resonance clusters requires the use of non self-adjoint operators, and second, the Stark perturbation is unbounded so control of the perturbation is achieved using localization properties of coherent states corresponding to hydrogen atom eigenvalues.